How can I plot a 3D phase space for a system of differential equations?

In summary, the conversation discusses how to plot a three-dimensional phase space for a set of differential equations in Mathematica. The summary suggests plotting the vector field defined by the right-hand side or typical solutions starting from various initial conditions. The conversation also mentions the possibility of using specialized toolboxes such as MatCont for further analysis. The conversation also touches upon the idea of using eigenvalues to determine the stability of critical points in the phase space.
  • #1
Aatifa
Hi, i would like to know how can i plot a three dimentionnal phase space (mathematica), for this kind of differential equations:
x'= (z^2(x-4y+z)+y^2(x+z)+z(x*y-x^2))/(z^2+x^2-2)
y'=y(y-3)+z(4y-z)+3(1-x^2)-2(y-3z)(z^2(x-4y+z)+y^2(x+z)+z(x*y-x^2))/x(z^2+x^2-2)
z'=2y-z(z^2(x-4y+z)+y^2(x+z)+z(x*y-x^2))/x(z^2+x^2-2)
do i need to solve numerically this system and then plot its solution?
 
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  • #2
The phase space itself would be ##\mathbb{R}^3## (or: a subset of ##\mathbb{R}^3##), so for that there is not much to plot. (Just three axes.)

You could plot the vector field defined by the right-hand side (I think Mathematica can do that?) or you could plot some "typical" solutions. For the former, you do not need to solve the ODE, for the latter you do need to solve it indeed. A combination is also possible: Plot the vector field and superimpose on this a plot of some typical solutions starting from various initial conditions.
 
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  • #3
Hi, thank you for your reply. Actually what i am trying to do is to plot the phase space of my dynamical system and superimpose on this plot the critical points of this system in order to study the stability of these critical points. Here is the result of the plot, unfortunatelly i didn't get any useful information from it.
I would like to know what did you mean by "typical" solutions?
 

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  • #4
Aatifa said:
Actually what i am trying to do is to plot the phase space of my dynamical system and superimpose on this plot the critical points of this system in order to study the stability of these critical points. Here is the result of the plot, unfortunatelly i didn't get any useful information from it.
Probably you already calculated the eigenvalues of the linearization at the critical points? Could you draw any conclusions regarding (local) stability from that?
Aatifa said:
I would like to know what did you mean by "typical" solutions?
I'm sorry, "typical" is indeed an ill-defined notion here. I meant to plot a couple of orbits through various initial conditions in the phase space to get an idea of what they look like. Likely this would work in Mathematica with some experimenting, but I am not familiar with that software. Alternatively, there are also more specialized (free) toolboxes that can help you with this and subsequent analysis. One of those is MatCont (software, Scholarpedia page), which is developed academically. (It requires MATLAB, though it might also run in Octave.)
 
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  • #5
Krylov said:
Probably you already calculated the eigenvalues of the linearization at the critical points? Could you draw any conclusions regarding (local) stability from that?

Yes, i did, actually i found some sadlle and stable points. But i was wondering if i could confirm this result by ploting the phase space.

I'm sorry, "typical" is indeed an ill-defined notion here. I meant to plot a couple of orbits through various initial conditions in the phase space to get an idea of what they look like. Likely this would work in Mathematica with some experimenting, but I am not familiar with that software. Alternatively, there are also more specialized (free) toolboxes that can help you with this and subsequent analysis. One of those is MatCont (software, Scholarpedia page), which is developed academically. (It requires MATLAB, though it might also run in Octave.)

thank you for your clarifications about my question, i will try to use one of these softwares that you have mensionned above.
 

What is a phase space plot?

A phase space plot is a graphical representation of the trajectory of a system's state over time, where the state is represented by multiple variables known as the system's phase space. It is commonly used in physics and engineering to analyze the behavior of dynamical systems.

How is a phase space plot created?

A phase space plot is created by plotting the values of multiple variables that describe the state of a system against each other on a graph. Each point on the graph represents a specific state of the system at a particular time. The trajectory of the system's state can then be visualized by connecting these points over time.

What can be learned from a phase space plot?

A phase space plot can provide insights into the behavior and stability of a system. It can help identify patterns, oscillations, and other characteristics of the system's state over time, which can be used to make predictions and improve the system's performance.

What are the benefits of using a phase space plot?

Phase space plots offer a visual representation of a system's behavior, which can be easier to interpret and analyze compared to other forms of data. They also allow for the identification of complex patterns and relationships between variables that may not be apparent from other types of data analysis.

Are there any limitations to using phase space plots?

One limitation of phase space plots is that they can only represent a limited number of variables at a time, which may not fully capture the complexity of some systems. Additionally, the accuracy and reliability of the plot depend on the quality of the data used to create it.

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